Optimal. Leaf size=69 \[ \frac{(a C+A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{a B-x (A c-a C)}{2 a c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.0431747, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1814, 12, 205} \[ \frac{(a C+A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{a B-x (A c-a C)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 12
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{\left (a+c x^2\right )^2} \, dx &=-\frac{a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}-\frac{\int \frac{-A-\frac{a C}{c}}{a+c x^2} \, dx}{2 a}\\ &=-\frac{a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}+\frac{(A c+a C) \int \frac{1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{a B-(A c-a C) x}{2 a c \left (a+c x^2\right )}+\frac{(A c+a C) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0531633, size = 68, normalized size = 0.99 \[ \frac{(a C+A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{-a B-a C x+A c x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 76, normalized size = 1.1 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ({\frac{ \left ( Ac-aC \right ) x}{2\,ac}}-{\frac{B}{2\,c}} \right ) }+{\frac{A}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77277, size = 412, normalized size = 5.97 \begin{align*} \left [-\frac{2 \, B a^{2} c +{\left (C a^{2} + A a c +{\left (C a c + A c^{2}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 2 \,{\left (C a^{2} c - A a c^{2}\right )} x}{4 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac{B a^{2} c -{\left (C a^{2} + A a c +{\left (C a c + A c^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (C a^{2} c - A a c^{2}\right )} x}{2 \,{\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.840179, size = 116, normalized size = 1.68 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c + C a\right ) \log{\left (- a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c + C a\right ) \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} - \frac{B a + x \left (- A c + C a\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15341, size = 81, normalized size = 1.17 \begin{align*} \frac{{\left (C a + A c\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} - \frac{C a x - A c x + B a}{2 \,{\left (c x^{2} + a\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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